7.1 Finite Difference-Taylor Formulae

8/10/2019 7.1 Finite Difference-Taylor Formulae

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UNIVERSITI TEKNOLOGI PETRONAS

PAB4233

ADVANCED RESERVOIR SIMULATION

JANUARY 2013

Principles of Finite Difference (1)

Petroleum Engineering Department (GPED)


2/23


3/23

Outline of the class

Principles of finite difference

Introduction

Overview of the big picture

Taylor series expansion

Discretization methods Reservoir Discretization

Discretization in spatial domain Constant grid block sizes

Variable grid block sizes

Grid Types

Time discretization

Time Derivatives

Numerical formulation

Explicit, Implicit, and Cranck-Nicholson


4/23

Introduction

The following exposition has two purposes:

(1) to define the terminology, and

(2) to summarize the basic facts which will be required later for thedevelopment of special techniques

The basic idea of any approximate method is to replace the original problem

by another problem that is easier to solve and whose solution is in some sense

close to the solution of the original problem


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Introduction, cont,

Definition

The numerical solution of partial differential equations by finite differences

refers to the process of replacing the partial derivatives by finite difference

quotients, and then obtaining solutions of the resulting system of algebraic

equations


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Three types of questions may be asked at this stage:

(a) How can a given differential equation be discretized?

(b) How can we ascertain that the finite-difference solution piis close to Pi in some sense,

and what is the magnitude of the difference?

(c) what is the best method of solving the resulting system of algebraic equations?

The first two questions are discussed in this lecture and the next one. The third

question is extremely important from the practical point of view, and involves two

steps.

First, whenever the finite-difference equations are nonlinear they must be linearized.

The second step involves the solution of resulting matrix equations, and this important

problem will be considered later

Introduction, cont,


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Finite-Difference Methods

Replace derivatives by differences

j j+1 j+2j-1j-2

1j 2jj

1j2j

1 jx 2 jxjx1 jx


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Overview

Overview of the computational solution procedures

Governing

Equations

ICs/BCs

DiscretizationSystem of

Algebraic

Equations

Equation

(Matrix)

Solver

Approximate

Solution

Continuous

Solutions

Finite-Difference

Finite-Volume

Finite-Element

Spectral

Boundary Element

Discrete

Nodal Values

Tridiagonal

ADI

SOR

Gauss-Seidel

Gaussian

elimination

Ui (x,y,z,t)

p (x,y,z,t)

T (x,y,z,t)


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Reservoir Simulation Process

Geological

Model

Simulation

Model


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m

o

m

o

mm

o

m

m

o

m

momom

m

o

oo

ooo

oooooo

xxm

xfxxaxf

mxfaxxaxxmmamxf

xfaaxf

xfaxxaaxf

xfaxxaxxaaxf

xfaxxaxxaxxaaxf

)(!

)()()(

!/)()(2)()1()!()(

!3/)(6)(

!2/)()(62)(

)()(3)(2)()()()()()(

0

)(

0

)(

1

)(

33

232

1

2

321

3

3

2

21

Taylor series expansion

Construction of finite-difference formula Numerical accuracy: discretization error

xo x


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Introduction to Finite Differences

A partial derivative replaced with a suitable algebraic difference quotient is called finite difference. Most

finite-difference representations of derivatives are based on Taylorsseries expansion.

Taylorsseries expansion:

Consider a continuous function of x, namely, f(x), with all derivatives defined at x. Then, the value of f at a

location can be estimated from a Taylor series expanded about point x, that is,

In general, to obtain more accuracy, additional higher-order terms must be included.

xx

...)(!1

...!3

1

!2

1

)()(3

3

32

2

2

n

n

n

xx

f

nxx

f

xx

f

xx

f

xfxxf

Any continuous differentiable function, in the vicinity of xi, can be expressed as a Taylor

series.


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Forward, Backward and Central Differences:

(1) Forward difference:

Neglecting higher-order terms, we can get

...)(

!

)(...)(

!3)(

!2

)()()()()( 1

3

33

1

2

22

111

in

nn

iii

iii

iiiiiii

x

f

n

xx

x

fxx

x

fxxxx

x

fxfxf

iii

i

ii

ii

iii xxx

x

xfxf

xx

xfxf

x

f

11

1

1

1

1 ;)()()()(

)(

(a)


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(2) Backward difference:

Neglecting higher-order terms, we can get

(3) Central difference:

(a)-(b) and neglecting higher-order terms, we can get

...)(

!

)()1(...)(

!3)(

!2

)()()()()( 1

3

33

1

2

22

111

in

nn

iin

iii

iii

iiiiix

f

n

xx

x

fxx

x

fxxxx

x

fxfxf

11

1

1 ;)()()()(

)(

iii

i

ii

ii

iii xxx

x

xfxf

xx

xfxf

x

f

(b)

11

11 )()()(

ii

iii

xx

xfxf

x

f

(c)


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(4) If , then (a), (b), (c) can be expressed as

Forward:

Backward:

Central:

Note:

xxx ii 1

x

ff

x

f iii

1)(

x

ff

x

f iii

1)(

x

ff

x

f iii

2)( 11

)(

)(

)(

11

11

ii

ii

ii

xff

xff

xff

(d)

(e)

(f)


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This is why we studied mass

conservation!!!


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Discretization

Discretization:

The word comes from discrete and is defined as constituting a separate thing;

individual; distinct; consisting of unconnected distinctparts.

Converts continuous PDE into difference form

Replaces original problem with other problem, which can be solved easily

The reservoir domain is presented by spatially distributed, interconnected discrete

elements (grid blocks)

Temporal (time) domain is also discretized

The reservoir parameters are calculated over these constitutive elements at discretetime steps


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Recap on previous slides

2

2

x

P

c

k

t

P

t

Diffusivity equation:

0

x

Pk

x Steady state:

Homogeneous: 02

2

x

P

x

P

x


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Discretization, Finite Difference Process

Divide the reservoir into numerous blocks and represents it with amesh of points or grid blocks.


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Discretization, cont,

Solve mathematical equations for each cell by numerical methods toobtain pressure, production and saturation changes with time.

The diffusivity equation (single phase, 1-D flow)

k2

2

x

P

c

k

t

P

t

Accuracy ofdata input

Impacts accuracy ofsimulator calculations

Finite Difference 3 Step Process


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t

P

k

c

x

P t

2

2

Continuous system: Discrete system:

Most flow problems encountered in reservoir engineering applications are too complex to be solved

by analytical methods.

Unlike analytic methods which give continuous solution in time and space (if an analytic solution

can be found), numerical approaches find solutions at discrete points in time and space.

The spatial domain is divided into a number of grids (also called cells or blocks) and the timedomain discretized to a number of time steps.

Continuous partial differential equation is then transformed to an equivalent discrete form of the

equation by finite-difference.


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The discretization of the differential equation yields a set of algebraic equations

whose solution gives an approximate response at discrete points in the domain.

The discretization technique most commonly used in reservoir simulation is the finite

difference method.

There are several ways of discretizing the fluid flow equations using finite difference

approximation. Common approaches are based on:

Taylor ser ies expansion,

variational approach, or

integral formulation


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Reservoir Discretization

Definition: The reservoir is described by a set of gridblocks (or gridpoints) whoseproperties, dimensions, boundaries, and locations in the reservoir are well defined.

The reservoir domain is presented by spatially distributed, interconnected discrete elements (grid

blocks)

Temporal (time) domain is also discretized

The reservoir parameters are calculated over these constitutive elements at discrete time steps

the grid system is defined with Nx gridblocks for a one dimensional model, with Nx

by Ny gridblocks for a two-dimensional model, and by by Nx Ny Nz for a three-

dimensional model.

The index is referred to the center and the unknowns such as pressure are calculated at

the center of a gridblock. This type of gridding systems is called the block centered

grid. The grid systems presented in the previous Figure have a uniform gridblock for

each of them.


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One dimensional grid system

Two dimensional grid system

Discretization in spatial domain

Three dimensional grid system

The grids in the numerical model are usually rectangular in form. Radial grids

are sometimes used in single-well modeling or local hybrid gridding system.

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7.1 Finite Difference-Taylor Formulae